TCG, or Tradeable Card Game, refers to an entire class of games involving trading cards that were created for the purpose of playing a game. Older games have often sometimes taken the moniker CCG instead, which can stand for Customizable or Collectible Card Game.
Traditional TCG's like Magic:The Gathering require the player to purchase packs of cards. In those packs will be a certain number of cards from the particular set, and in a certain distribution of rarity. The game designers will assign each card as either "common", "uncommon", or "rare". In theory, rare cards should be more desireable, and so they are naturally made more difficult to acquire. This is strictly a capitalist move that card companies write off with the excuse of being able to trade cards with other people to get the ones that you need.
Using the player's collection, they construct a deck out of any of the cards they have available. Most games have deckbuilding rules, and most frequently expect 60 cards in the player's deck. Each player's deck may be very different from their opponents, and this is one of the hallmarks of TCG gaming. Creative or ingenious players may often be well-poised to win before the game has even begun.
Different games play with very different overall rules, goals, and strategies. Some of the more well-known TCGs include Magic: The Gathering, Yu-gi-oh!, Vs., Pokemon, and World of Warcraft.
In more recent times, some game designers have come up with twists on the classic business model. Eye of Judgement was a game released for the Playstation 3 that uses tradeable cards as a TCG would, but also incorporates electronic elements to the game to keep track of extra variables and to allow play with people in other parts of the world.
Everquest developers have also licensed a game, Legends of Norrath, that is both played on-line and that awards players of Everquest with actual packs of cards.